# Helmholtz reciprocity

The Helmholtz reciprocity principle describes how a ray of light and its reverse ray encounter matched optical adventures, such as reflections, refractions, and absorptions in a passive medium, or at an interface. It does not apply to moving, non-linear, or magnetic media.

For example, incoming and outgoing light can be considered as reversals of each other,[1] without affecting the bidirectional reflectance distribution function (BRDF)[2] outcome. If light was measured with a sensor and that light reflected on a material with a BRDF that obeys the Helmholtz reciprocity principle one would be able to swap the sensor and light source and the measurement of flux would remain equal.

In the computer graphics scheme of global illumination, the Helmholtz reciprocity principle is important if the global illumination algorithm reverses light paths (for example Raytracing versus classic light path tracing).

## Physics

The Stokes-Helmholtz reversion-reciprocity principle[3][4][5][6][7][8][9][10][11][12][13][1][14][15][16][17][18][19][20][21][22] was stated in part by Stokes (1849)[3] and with reference to polarization on page 169 [4] of Helmholtz's Handbuch der physiologischen Optik of 1856 as cited by Kirchhoff[8] and by Planck.[13]

As cited by Kirchhoff in 1860, the principle is translated as follows:

A ray of light proceeding from point 1 arrives at point 2 after suffering any number of refractions, reflections, &c. At point 1 let any two perpendicular planes a1, b1 be taken in the direction of the ray; and let the vibrations of the ray be divided into two parts, one in each of these planes. Take similar planes a2, b2 in the ray at point 2; then the following proposition may be demonstrated. If when the quantity of light i polarized in the plane a1 proceeds from 1 in the direction of the given ray, that part k thereof of light polarized in a2 arrives at 2, then, conversely, if the quantity of light i polarized in a2 proceeds from 2, the same quantity of light k polarized in a1 [Kirchhoff's published text here corrected by Wikipedia editor to agree with Helmholtz's 1867 text] will arrive at 1.[8]

The most extremely simple statement of the principle is 'if I can see you, then you can see me'. Rayleigh stated the basic idea of reciprocity as a consequence of the linearity of propagation of small vibrations, light consisting of sinusoidal vibrations in a linear medium.[9][10][11][12]

Like the principles of thermodynamics, this principle is reliable enough to use as a check on the correct performance of experiments, in contrast with the usual situation in which the experiments are tests of a proposed law.[1][12]

In his magisterial proof[23] of the validity of Kirchhoff's law of equality of radiative emissivity and absorptivity,[24] Planck makes repeated and essential use of the Stokes-Helmholtz reciprocity principle.

When there are magnetic fields in the path of the ray, the principle does not apply.[4] Departure of the optical medium from linearity causes departure from Helmholtz reciprocity. When there are moving objects in the path of the ray, the principle may be entirely inapplicable.

Helmholtz reciprocity referred originally to light. This is a particular form of electromagnetism that may be called far-field radiation. For this, the electric and magnetic fields do not need distinct descriptions, because they propagate feeding each other evenly. So the Helmholtz principle is a more simply described special case of electromagnetic reciprocity in general, which is described by distinct accounts of the interacting electric and magnetic fields. The Helmholtz principle rests mainly on the linearity and superposability of the light field, and it has close analogues in non-electromagnetic linear propagating fields, such as sound. It was discovered before the electromagnetic nature of light became known.[9][10][11][12]

## References

1. Hapke, B. (1993). Theory of Reflectance and Emittance Spectroscopy, Cambridge University Press, Cambridge UK, ISBN 0-521-30789-9, Section 10C, pages 263-264.
2. Hapke, B. (1993). Theory of Reflectance and Emittance Spectroscopy, Cambridge University Press, Cambridge UK, ISBN 0-521-30789-9, Chapters 8-9, pages 181-260.
3. Stokes, G.G. (1849). On the perfect blackness of the central spot in Newton's rings, and on the verification of Fresnel's formulae for the intensities of reflected and refracted rays, Cambridge and Dublin Mathematical Journal, new series, 4: 1-14.
4. Helmholtz, H. von (1856). Handbuch der physiologischen Optik, first edition cited by Planck, Leopold Voss, Leipzig, volume 1, page 169.[1]
5. Helmholtz, H. von (1903). Vorlesungen über Theorie der Wärme, edited by F. Richarz, Johann Ambrosius Barth, Leipzig, pages 158-162.
6. Helmholtz, H. (1859/60). Theorie der Luftschwingungen in Röhren mit offenen Enden, Crelle's Journal für die reine und angewandte Mathematik 57(1): 1-72, page 29.
7. Stewart, B. (1858). An account of some experiments on radiant heat, involving an extension of Professor Prevost's theory of exchanges, Trans. Roy. Soc. Edinburgh 22 (1): 1-20, page 18.
8. Kirchhoff, G. (1860). On the Relation between the Radiating and Absorbing Powers of different Bodies for Light and Heat, Ann. Phys., 119: 275-301, at page 287 [2], translated by F. Guthrie, Phil. Mag. Series 4, 20:2-21, at page 9.
9. Strutt, J.W. (Lord Rayleigh) (1873). Some general theorems relating to vibrations, Proc. Lond. Math. Soc. 4: 357-368, pages 366-368.
10. Rayleigh, Lord (1876). On the application of the Principle of Reciprocity to acoustics, Proc. Roy. Soc. A, 25: 118-122.
11. Strutt, J.W., Baron Rayleigh (1894/1945). The Theory of Sound, second revised edition, Dover, New York, volume 1, sections 107-111a.
12. Rayleigh, Lord (1900). On the law of reciprocity in diffuse reflection, Phil. Mag. series 5, 49: 324-325.
13. Planck, M. (1914). The Theory of Heat Radiation, second edition translated by M. Masius, P. Blakiston's Son and Co., Philadelphia, page 35.
14. Minnaert, M. (1941). The reciprocity principle in lunar photometry, Astrophysical Journal 93: 403-410.[3]
15. Mahan, A.I. (1943). A mathematical proof of Stokes' reversibility principle, J. Opt. Soc. Am., 33(11): 621-626.
16. Chandrasekhar, S. (1950). Radiative Transfer, Oxford University Press, Oxford, pages 20-21, 171-177, 182.
17. Tingwaldt, C.P. (1952). Über das Helmholtzsche Reziprozitätsgesetz in der Optik, Optik, 9(6): 248-253.
18. Levi, L. (1968). Applied Optics: A Guide to Optical System Design, 2 volumes, Wiley, New York, volume 1, page 84.
19. Clarke, F.J.J., Parry, D.J. (1985). Helmholtz reciprocity: its validity and application to reflectometry, Lighting Research & Technology, 17(1): 1-11.
20. Lekner, J. (1987). Theory of reflection, Martinus Nijhoff, Dordrecht, ISBN 90-247-3418-5, pages 33-37.[4]
21. Born, M., Wolf, E. (1999). Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th edition, Cambridge University Press, ISBN 0-521-64222-1, page 423.
22. Potton, R.J. (2004), Reciprocity in optics, Rep. on Prog. Phys. 76: 717-754 [5].
23. Planck, M. (1914). The Theory of Heat Radiation, second edition translated by M. Masius, P. Blakiston's Son and Co., Philadelphia, pages 35, 38,39.
24. Kirchhoff, G. (1860). On the Relation between the Radiating and Absorbing Powers of different Bodies for Light and Heat, Ann. Phys., 119: 275-301 [6], translated by F. Guthrie, Phil. Mag. Series 4, 20:2-21.